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Most philosophers believe that, other things being equal, simpler theories are better. But what exactly does theoretical simplicity amount to? Syntactic simplicity, or elegance, measures the number and conciseness of the theory's basic principles. Ontological simplicity, or parsimony, measures the number of kinds of entities postulated by the theory. One issue concerns how these two forms of simplicity relate to one another. There is also an issue concerning the justification of principles, such as Occam's Razor, which favor simple theories. The history of philosophy has seen many approaches to defending Occam's Razor, from the theological justifications of the Early Modern period, to contemporary justifications employing results from probability theory and statistics. *1. Introduction *2. Ontological Parsimony *3. A Priori Justifications of Simplicity *4. Naturalistic Justifications of Simplicity *5. Probabilistic/Statistical Justifications of Simplicity *6. Other Issues Concerning Simplicity **6.1 Quantitative Parsimony **6.2 Principles of Plenitude **6.3 Simplicity and Induction *Bibliography *Academic Tools *Other Internet Resources *Related Entries ---- 1. Introduction There is a widespread philosophical presumption that simplicity is a theoretical virtue. This presumption that simpler theories are preferable appears in many guises. Often it remains implicit; sometimes it is invoked as a primitive, self-evident proposition; other times it is elevated to the status of a ‘Principle’ and labeled as such (for example, the ‘Principle of Parsimony’). However, it is perhaps best known by the name ‘Occam's (or Ockham's) Razor.’ Simplicity principles have been proposed in various forms by theologians, philosophers, and scientists, from ancient through medieval to modern times. Thus Aristotle writes in his Posterior Analytics, We may assume the superiority ceteris paribus of the demonstration which derives from fewer postulates or hypotheses.http://plato.stanford.edu/entries/simplicity/notes.html#1 1 Moving to the medieval period, Aquinas writes If a thing can be done adequately by means of one, it is superfluous to do it by means of several; for we observe that nature does not employ two instruments where one suffices (Aquinas 1945, p. 129). Kant—in the Critique of Pure Reason—supports the maxim that “rudiments or principles must not be unnecessarily multiplied (entia praeter necessitatem non esse multiplicanda)” and argues that this is a regulative idea of pure reason which underlies scientists' theorizing about nature (Kant 1950, pp. 538–9). Both Galileo and Newton accepted versions of Occam's Razor. Indeed Newton includes a principle of parsimony as one of his three ‘Rules of Reasoning in Philosophy’ at the beginning of Book III of Principia Mathematica. Rule I: We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances. Newton goes on to remark that “Nature is pleased with simplicity, and affects not the pomp of superfluous causes” (Newton 1972, p. 398). Galileo, in the course of making a detailed comparison of the Ptolemaic and Copernican models of the solar system, maintains that “Nature does not multiply things unnecessarily; that she makes use of the easiest and simplest means for producing her effects; that she does nothing in vain, and the like” (Galileo 1962, p. 397). Nor are scientific advocates of simplicity principles restricted to the ranks of physicists and astronomers. Here is the chemist Lavoisier writing in the late 18th Century If all of chemistry can be explained in a satisfactory manner without the help of phlogiston, that is enough to render it infinitely likely that the principle does not exist, that it is a hypothetical substance, a gratuitous supposition. It is, after all, a principle of logic not to multiply entities unnecessarily (Lavoisier 1862, pp. 623–4). Compare this to the following passage from Einstein, writing 150 years later. The grand aim of all science…is to cover the greatest possible number of empirical facts by logical deductions from the smallest possible number of hypotheses or axioms (Einstein, quoted in Nash 1963, p. 173). Editors of a recent volume on simplicity sent out surveys to 25 recent Nobel laureates in economics. Almost all replied that simplicity played a role in their research, and that simplicity is a desirable feature of economic theories (Zellner et al. 2001, p.2). Within philosophy, Occam's Razor (OR) is often wielded against metaphysical theories which involve allegedly superfluous ontological apparatus. Thus materialists about the mind may use OR against dualism, on the grounds that dualism postulates an extra ontological category for mental phenomena. Similarly, nominalists about abstract objects may use OR against their platonist opponents, taking them to task for committing to an uncountably vast realm of abstract mathematical entities. The aim of appeals to simplicity in such contexts seem to be more about shifting the burden of proof, and less about refuting the less simple theory outright. The philosophical issues surrounding the notion of simplicity are numerous and somewhat tangled. The topic has been studied in piecemeal fashion by scientists, philosophers, and statisticians. The apparent familiarity of the notion of simplicity means that it is often left unanalyzed, while its vagueness and multiplicity of meanings contributes to the challenge of pinning the notion down precisely.http://plato.stanford.edu/entries/simplicity/notes.html#2 2 A distinction is often made between two fundamentally distinct senses of simplicity: syntactic simplicity (roughly, the number and complexity of hypotheses), and ontological simplicity (roughly, the number and complexity of things postulated).http://plato.stanford.edu/entries/simplicity/notes.html#3 3 These two facets of simplicity are often referred to as elegance and parsimony respectively. For the purposes of the present overview we shall follow this usage and reserve ‘parsimony’ specifically for simplicity in the ontological sense. It should be noted, however, that the terms ‘parsimony’ and ‘simplicity’ are used virtually interchangeably in much of the philosophical literature. Philosophical interest in these two notions of simplicity may be organized around answers to three basic questions; (i) How is simplicity to be defined? Definition (ii) What is the role of simplicity principles in different areas of inquiry? Usage (iii) Is there a rational justification for such simplicity principles? Justification As we shall see, answering the definitional question, (i), is more straightforward for parsimony than for elegance. Conversely, more progress on the issue, (iii), of rational justification has been made for elegance than for parsimony. It should also be noted that the above questions can be raised for simplicity principles both within philosophy itself and in application to other areas of theorizing, especially empirical science. With respect to question (ii), there is an important distinction to be made between two sorts of simplicity principle. Occam's Razor may be formulated as an epistemic principle: if theory T is simpler than theory T*, then it is rational (other things being equal) to believe T rather than T*. Or it may be formulated as a methodological principle: if T is simpler than T* then it is rational to adoptT as one's working theory for scientific purposes. These two conceptions of Occam's Razor require different sorts of justification in answer to question (iii). In analyzing simplicity, it can be difficult to keep its two facets—elegance and parsimony—apart. Principles such as Occam's Razor are frequently stated in a way which is ambiguous between the two notions, for example, “Don't multiply postulations beyond necessity.” Here it is unclear whether ‘postulation’ refers to the entities being postulated, or the hypotheses which are doing the postulating, or both. The first reading corresponds to parsimony, the second to elegance. Examples of both sorts of simplicity principle can be found in the quotations given earlier in this section. While these two facets of simplicity are frequently conflated, it is important to treat them as distinct. One reason for doing so is that considerations of parsimony and of elegance typically pull in different directions. Postulating extra entities may allow a theory to be formulated more simply, while reducing the ontology of a theory may only be possible at the price of making it syntactically more complex. For example the postulation of Neptune, at the time not directly observable, allowed the perturbations in the orbits of other observed planets to be explained without complicating the laws of celestial mechanics. There is typically a trade-off between ontology and ideology—to use the terminology favored by Quine—in which contraction in one domain requires expansion in the other. This points to another way of characterizing the elegance/parsimony distinction, in terms of simplicity of theory versus simplicity of world respectively.http://plato.stanford.edu/entries/simplicity/notes.html#4 4 Sober 2001 argues that both these facets of simplicity can be interpreted in terms of minimization. In the (atypical) case of theoretically idle entities, both forms of minimization pull in the same direction; postulating the existence of such entities makes both our theories (of the world) and the world (as represented by our theories) less simple than they might be. 2. Ontological Parsimony Perhaps the most common formulation of the ontological form of Occam's Razor is the following: (OR) Entities are not to be multiplied beyond necessity. It should be noted that modern formulations of Occam's Razor are connected only very tenuously to the 14th-century figure William of Ockham. We are not here interested in the exegetical question of how Ockham intended his ‘Razor’ to function, nor in the uses to which it was put in the context of medieval metaphysics.http://plato.stanford.edu/entries/simplicity/notes.html#5 5 Contemporary philosophers have tended to reinterpret OR as a principle of theory choice: OR implies that—other things being equal—it is rational to prefer theories which commit us to smaller ontologies. This suggests the following paraphrase of OR: (OR1) Other things being equal, if T1 is more ontologically parsimonious than T2 then it is rational to prefer T1 to T2. What does it mean to say that one theory is more ontologically parsimonious than another? The basic notion of ontological parsimony is quite straightforward, and is standardly cashed out in terms of Quine's concept of ontological commitment. A theory, T, is ontologically committed to Fs if and only if T entails that F's exist (Quine 1981, pp. 144–4). If two theories, T1 and T2, have the same ontological commitments except that T2 is ontologically committed to Fs and T1 is not, then T1 is more parsimonious than T2. More generally, a sufficient condition for T1 being more parsimonious than T2 is for the ontological commitments of T1 to be a proper subset of those of T2. Note that OR1is considerably weaker than the informal version of Occam's Razor, OR, with which we started. OR stipulates only that entities should not be multiplied beyond necessity. OR1, by contrast, states that entities should not be multiplied other things being equal, and this is compatible with parsimony being a comparatively weak theoretical virtue. One ‘easy’ case where OR1 can be straightforwardly applied is when a theory, T, postulates entities which are explanatorily idle. Excising these entities from T produces a second theory, T*, which has the same theoretical virtues as T but a smaller set of ontological commitments. Hence, according to OR1, it is rational to pick T* over T. (As previously noted, terminology such as ‘pick’ and ‘prefer’ is crucially ambiguous between epistemic and methodological versions of Occam's Razor. For the purposes of defining ontological parsimony, it is not necessary to resolve this ambiguity.) However, such cases are presumably rare, and this points to a more general worry concerning the narrowness of application of OR1. First, how often does it actually happen that we have two (or more) competing theories for which ‘other things are equal’? As biologist Kent Holsinger remarks, Since Occam's Razor ought to be invoked only when several hypotheses explain the same set of facts equally well, in practice its domain will be very limited…Cases where competing hypotheses explain a phenomenon equally well are comparatively rare (Holsinger 1980, pp. 144–5). Second, how often are one candidate theory's ontological commitments a proper subset of another's? Much more common are situations where ontologies of competing theories overlap, but each theory has postulates which are not made by the other. Straightforward comparisons of ontological parsimony are not possible in such cases. Before setting aside the definitional question for ontological parsimony, one further distinction should be mentioned. This distinction is between qualitative parsimony (roughly, the number of types (or kinds) of thing postulated) and quantitative parsimony (roughly, the number of individual things postulated).http://plato.stanford.edu/entries/simplicity/notes.html#6 6 The default reading of Occam's Razor in the bulk of the philosophical literature is as a principle of qualitative parsimony. Thus Cartesian dualism, for example, is less qualitatively parsimonious than materialism because it is committed to two broad kinds of entity (mental and physical) rather than one. Section 6.1 contains a brief discussion of quantitative parsimony; apart from this the focus will be on the qualitative notion. It should be noted that interpreting Occam's Razor in terms of kinds of entity brings with it some extra philosophical baggage of its own. In particular, judgments of parsimony become dependent on how the world is sliced up into kinds. Nor is guidance from extra-philosophical usage—and in particular from science—always clearcut. For example, is a previously undiscovered subatomic particle made up of a novel rearrangement of already discovered sub-particles a new ‘kind’? What about a biological species, which presumably does not contain any novel basic constituents? Also, ought more weight to be given to broad and seemingly fundamental divisions of kind—for example between the mental and physical—than between more parochial divisions? Intuitively, the postulation of a new kind of matter would seem to require much more extensive and solid justification than the postulation of a new sub-species of spider.http://plato.stanford.edu/entries/simplicity/notes.html#7 7 The third and final question from Section 1 concerns potential justifications for principles of ontological parsimony such as Occam's Razor. The demand for justification of such principles can be understood in two importantly distinct ways, corresponding to the distinction between epistemic principles and methodological principles made at the end of Section 1. Justifying an epistemic principle requires answering an epistemic question: why are parsimonious theories more likely to be true? Justifying a methodological principle requires answering a pragmatic question: why does it make practical sense for theorists to adopt parsimonious theories?http://plato.stanford.edu/entries/simplicity/notes.html#8 8 Most attention in the literature has centered on the first, epistemic question. It is easy to see how syntactic elegance in a theory can bring with it pragmatic advantages such as being more perspicuous, being easier to use and manipulate, and so on. But the case is more difficult to make for ontological parsimony.http://plato.stanford.edu/entries/simplicity/notes.html#9 9 It is unclear what particular pragmatic disadvantages accrue to theories which postulate extra kinds of entities; indeed—as was mentioned in the previous section—such postulations can often bring with them striking syntactic simplification. Before looking at approaches to answering the epistemic justification question, mention should be made of two positions in the literature which do not fall squarely into either the pragmatic or epistemic camp. The first position, associated primarily with Quine, argues that parsimony carries with it pragmatic advantages and that pragmatic considerations themselves provide rational grounds for discriminating between competing theories (Quine 1966, Walsh 1979). The Quinean position bases an answer to the second question on the answer to the first, thus blurring the boundary between pragmatic and epistemic justification. The second position, due to Sober, rejects the implicit assumption in both the above questions that some global justification of parsimony can be found (Sober 1988, 1994). Instead Sober argues that appeals to parsimony always depend on local background assumptions for their rational justification. Thus Sober writes: The legitimacy of parsimony stands or falls, in a particular research context, on subject matter specific (and a posteriori) considerations. … What makes parsimony reasonable in one context may have nothing in common with why it matters in another (Sober 1994). Philosophers who reject these arguments of Quine and Sober, and thus take the demand for a global, epistemic justification seriously, have developed a variety of approaches to justifying parsimony. Most of these approaches can be collected under two broad headings: (A) A priori philosophical, metaphysical, or theological justifications. (B) Naturalistic justifications, based on appeal to scientific practice. As we shall see, the contrast between these two sorts of approach mirrors a broader divide between the rival traditions of rationalism and empiricism in philosophy as a whole. As well as parsimony, the question of rational justification can also be raised for principles based on elegance, the second facet of simplicity distinguished in Section 1. Approaches to justifying elegance along the lines of (A) and (B) are possible, but much of the recent work falls under a third category; © Justifications based on results from probability theory and/or statistics. The next three sections examine these three modes of justification of simplicity principles. The a priori justifications in category (A) concern simplicity in both its parsimony and elegance forms. The justifications falling under category (B) pertain mostly to parsimony, while those falling under category © pertain mostly to elegance. 3. A Priori Justifications of Simplicity The role of simplicity as a theoretical virtue seems so widespread, fundamental, and implicit that many philosophers, scientists, and theologians have sought a justification for principles such as Occam's Razor on similarly broad and basic grounds. This rationalist approach is connected to the view that making a priori simplicity assumptions is the only way to get around the underdetermination of theory by data. Until the second half of the 20th Century this was probably the predominant approach to the issue of simplicity. More recently, the rise of empiricism within analytic philosophy led many philosophers to argue disparagingly that a priori justifications keep simplicity in the realm of metaphysics (see Zellner et al. 2001, p.1). Despite its changing fortunes, the rationalist approach to simplicity still has its adherents. For example, Richard Swinburne writes: I seek…to show that—other things being equal—the simplest hypothesis proposed as an explanation of phenomena is more likely to be the true one than is any other available hypothesis, that its predictions are more likely to be true than those of any other available hypothesis, and that it is an ultimate a priori epistemic principle that simplicity is evidence for truth (Swinburne 1997, p. 1).